Greetings, friends. I’m Timur Garanin. For those who are not familiar with me, I am engaged in the creation of educational materials
in physics, electronics and radio engineering.
My YouTube channel already has a lot of videos about antennas, including fractal antennas.
The public interest in this type of antenna does not fade away, on the contrary, it is growing. And this is not surprising.
Fractal antennas combine such qualities as compactness, broadband and satisfactory gain.
An outstanding set of parameters. Nevertheless, today fractal antennas exist almost as a phenomenon.
The mathematical apparatus is either completely absent or contained in rare scientific papers and is complicated to the point of absurdity.
You can find many works, including amateur ones, on the study of the frequency properties of fractal antennas.
But these properties are analyzed either after the fact, by the finished antenna, or by computer simulation.
As such, there is practically no algorithm for calculating the frequency properties of fractal antennas.
In the end, it's time to deal with this class of antennas, and create a more or less transparent algorithm for calculating them.
This topic requires serious study.
Do I have the competency to do this?
For the last three months, I’ve been doing raking of fractal blockages,
systematization of all information about the properties of fractal structures in iron. As a result, a new book was born -
"Fractal antennas and delay lines"
And I have good news for you.
The mathematical apparatus is possible and exists.
Moreover, ordered by me
to achieve maximum transparency.
But we will come to this a little later. Now let's go through the book.
The book is presented in pdf format and has an interactive table of contents,
which with such a volume of material is simply necessary.
We start with the simplest.
We understand what fractal antennas are, find out that they have a constant dimension and a known number of iterations,
which makes it possible to create a mathematical apparatus.
Next, we indicate the advantages and disadvantages of fractal antennas.
And we list the main areas in which fractal antennas are used.
In the chapter "The principle of operation of fractal antennas" we understand where they come from several resonances, in contrast to linear antennas with one resonance.
In the next chapter, we look at the most typical elementary structures,
which are emitters in fractal antennas. And also, what influence do the elements of intermediate orders have.
In the chapter on the radiation patterns of fractal antennas, we understand
why their DNs have such a complex shape, such a strong frequency dependence,
and is there any sense at all in applying the usual approach to the analysis of DNs here.
This is followed by an extremely important chapter in which we analyze the frequency properties of fractal antennas.
We find that the resonances of fractal antennas differ from the resonances of linear antennas
and with the same length of the canvas, and with the same dimensions.
We find out that this is due to the fact that, from a mathematical point of view, the dimension of the fractal curve is greater than unity,
therefore, the calculation as for linear antennas is not appropriate here. From a physical point of view, this is due to the presence of a huge number of capacitive (as well as inductive) connections,
as a result, the signal in the canvas of the fractal antenna propagates faster than in the canvas of the linear antenna.
We used to think that in a linear antenna
the signal travels at the speed of light, adjusted for a factor of shortening.
So, in the canvas of fractal antennas, the signal propagates much faster.
We show that when trying to computer simulate and determine resonant frequencies
We often face discrepancies with experimental data.
We find out what explains this.
We consider examples of resonances of fractal elements. Using the first iteration of a Koch fractal in a loop antenna as an example
show where the second resonance came from.
We draw preliminary conclusions regarding the broadband and gain of fractal antennas.
In the next subsection, we discuss such an amazing opportunity as manipulating the resonances of a fractal antenna.
We find that by changing the dimension of the fractal and the number of iterations,
we can not only move the resonant frequencies, but also shift and move them away from each other.
Fractal approach gives us unprecedented opportunities
for managing the frequency properties of the antenna. We can theoretically create any picture of the resonances of the antenna.
Moreover, no one bothers us to combine fractal vibrators into gratings
with complementary frequency properties, and receive amplification in the entire band without dips.
The next chapter is devoted to
gain factor
fractal antennas.
We understand the objective reasons for the deterioration of gain -
a decrease in the effective length of the antenna and the appearance of countercurrents, along with the distortion of polarization and radiation pattern.
In the chapter "resistance and matching of fractal antennas"
we find that resistance decreases with increasing iterations.
This does not matter, because antenna resistance can be adjusted
shifting the connection point to the feeder. But this simultaneously changes the operating mode of the antenna.
Moreover, matching the antenna over the entire range is impossible in principle.
And then the most important chapter awaits us, for the sake of which it is at least worth studying this book.
We analyze the mathematical apparatus,
allowing to calculate the frequency properties of fractal antennas.
We determine the fractal dimension, the number of iterations, and calculate the frequency coefficient.
We calculate the frequency of the linear antenna subjected to folding into a fractal, find its wavelength and a quarter of its length.
Determine the size of the resulting fractal antenna after folding. We analyze all this with specific examples.
Moreover, we calculate the frequencies of higher resonances.
There is no such apparatus anywhere in the finished state. Only this book provides such a tool.
In the next chapter, we consider the algorithm
algorithm for creating a fractal antenna from a linear one.
We carry out the already known calculations, determine the dimension, the frequency coefficient, calculate the antenna frequency before folding and its length.
We perform these operations for the antennas of the first and second iteration of the Koch square.
The next chapter is about creating loop antennas.
First, we create a conventional loop antenna at a given frequency.
Then, after already familiar calculations, we create a loop antenna of the first iteration of the Koch square, and in addition to the fundamental frequency we find the frequency of the second resonance.
After that, we create a loop antenna at a given frequency for the second iteration of the Koch square.
For it, we already find two higher resonances tied to its internal structures.
As well as the resulting dimension of the antenna and the length of its canvas.
Showing
that segmentation of a fractal antenna removes the lowest frequency resonance, while maintaining the highest.
Next, we consider what advantages fractal antennas give us on the surface of printed circuit boards.
We show that the most profitable options are curves filling the space, and patch gratings,
since it is precisely these types of antennas that use the board resource most efficiently.
In the chapter “tree-shaped fractal antennas” we consider what advantages fractal trees give us.
Ready-made power dividers, matching of all branches and efficient filling of the space.
Examples where tree feeders are already used. What types of trees exist and what are their features.
Next, we consider how slotted fractal antennas work.
We study single emitters, tree structures, and gratings with emitters of various shapes.
Using the Sierpinski napkin as an example, we determine
that self-similar triangular elements work as elementary emitters of the array, forming many resonances.
In terms of space utilization, the most effective structures are Peano curves.
We consider the features of the Hilbert curve, an algorithm for constructing a U-shaped curve on a square plane, and a line transformation algorithm.
Also, methods for creating a loop antenna based on a Hilbert P-curve.
As a second example, we consider the S-curve.
Algorithm for constructing a fractal with several ways of alternating S-elements,
and characteristics of the resulting curves.
We study the options for constructing loop antennas in the main ways and combined.
We compare the characteristics of the considered curves.
We compare examples of creating dipoles on square planes, and the features of converting lines with two fractals.
Next, we come to the study of fractal delay lines. First, remember what the delay lines are all about,
why are they needed, what types are there, and what are their features.
We construct the fractal delay line based on the Hilbert curve and rightly point out its advantages.
Next, we construct the delay lines based on the S-shaped curve, in three versions, depending on the position and type of alternation of S-elements.
We discuss the possibility of management
characteristics of fractal delay lines by manipulating their dimensions, the number of iterations, the alternation and position of their elements.
Then we move on to the most known structure in the delay lines - the meander.
We study what elements it consists of, and what affects its linear parameters.
How to manipulate the form factor and dimensions to achieve the specified parameters.
Examples of single-wire and two-wire meanders.
Since we have already dealt with fractal vibrators, and with tree structures, and with methods of packing delay lines,
then nothing prevents us from using all this to build antenna arrays.
To begin with, we create power dividers from tree-shaped fractals, getting a ready-made coordinated parallel feeder structure that can work in the entire frequency range.
Next, we use fractal delay lines to seal the gratings with the sequential inclusion of vibrators.
Using the property of fractal structures to work in antenna array mode,
we create single and multi-band structures from flat triangular and rectangular vibrators.
We are exploring our capabilities for maximizing the use of PCB resources,
and the ability to create wideband slotted gratings.
The final chapter is devoted to the practical manufacture of fractal antennas. First, we paint the algorithm for creating a simple wire fractal antenna.
After that, the algorithm for manufacturing antennas on the board. And then we specify the sequence of actions,
which will allow to achieve the best results with the least expenditure of resources in modeling and full-scale tuning of antennas and antenna arrays.
The book ends with a conclusion in which we summarize the possibilities
which open fractal algorithms for us in building antenna systems.
And these opportunities are unprecedented.
By changing the dimensions of fractals at different iterations,
by shifting and separating the resonant frequencies, we can flexibly form the working bands of the antennas in any frequency range and any width.
Well, that’s what material I’ve been working on for the last three months.
Despite the fact that the book has already turned out to be voluminous, I must rightly note
that research in this direction is at an early stage. We are waiting for a real evolutionary leap in physics and mathematics,
when we fully learn to use the fractal approach.
I highly recommend reading the book
firstly to radio technicians, and secondly to all experts in the exact sciences,
mathematicians, physicists, programmers. Go ahead.
Count, create, explore.
This is a very interesting and promising topic. A completely new class of antennas,
more precisely the ability to control frequency properties. This is where I end. Study the book. Wish you luck!